In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra whose generalized Cartan matrix is positive semi-definite and has corank 1. From purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite dimensional, semisimple Lie algebras is much better understood than that of general (hyperbolic) Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and conformal field theory due to the way they are constructed: starting from a simple Lie algebra <math>\mathfrak{g}</math>, one considers the loop algebra, <math>L\mathfrak{g}</math>, formed by the <math>\mathfrak{g}</math>-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra <math>\hat{\mathfrak{g}}</math> is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physists call a quantum anomaly. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations are given by modular forms.
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Affine Lie algebras from simple Lie algebras
Construction
Each (untwisted) affine Lie algebra may be constructed from a finite-dimensional semi-simple Lie algebra. Semi-simple Lie algebras are direct sums of commuting simple Lie algebras, and the corresponding affine Lie algebras are also direct sums of the affine versions of each simple Lie algebra. Thus it will suffice to consider affine Lie algebras constructed from simple Lie algebras. The way in which we construct the affine Lie algebra <math>\hat{\mathfrak{g}}</math> associated to a finite dimensional simple Lie algebra <math>\mathfrak{g}</math> goes as follows. <math>\hat{\mathfrak{g}}</math> is defined to be the central extension
Where <math>\mathbb{C}[t,t^{-1}]</math> denotes the set of all Laurent polynomials in the indeterminate t. The bracket is defined by
for all <math>a,b\in\mathfrak{g}</math> and <math>n,m\in\mathbb{Z}</math>, where <math>\langle\cdot |\cdot\rangle</math> is the usual Cartan-Killing form on <math>\mathfrak{g}</math> and <math>c</math> is the basis element of the central extension. Note that the bracket on the left side of the definition is the "new" bracket being defined on <math>\tilde{\mathfrak{g}}</math> and the bracket on the right side is the "old" bracket of <math>\mathfrak{g}</math>. The algebra <math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]</math> is sometimes called the loop algebra associated to <math>\mathfrak{g}</math>.
Constructing the Dynkin diagrams
The Dynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root. Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group of outer automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras.
Classifying the central extensions
The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as a central extension of the loop algebra of the corresponding simple Lie algebra. If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra. In physics, one often considers instead the direct sum of a semisimple algebra and an abelian algebra <math>\mathbb{C}^n</math>. In this case one also needs to add n further central elements for the n abelian generators. The second integral cohomology of the loop group of the corresponding simple compact Lie group is isomorphic to the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the first Chern class of the fibration. Therefore the central extensions of an affine Lie group are classified by a single parameter k which is called the central charge in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist when k is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge for each simple component.
Noncompact real forms
Of course the number of central charges does not depend on the real form that is chosen, as central charges are defined already for the complex Lie algebra, however different choices of real form lead to different unitary representations. If one chooses a noncompact real form one finds a much richer classification unitary representations, including some continuous families in which the central charge is not integral.
Applications
They appear naturally in theoretical physics (for example, in conformal field theories such as the WZW model and coset models and even on the worldsheet of the heterotic string), geometry, and elsewhere in mathematics.
References
- V. Kac, Infinite dimensional Lie algebras, ISBN 0-521-37215-1
- Jurgen Fuchs, Affine Lie Algebras and Quantum Groups, (1992) Cambridge University Press. ISBN 0-521-48412-X
- Toshitake Kohno, Conformal Field Theory and Topology, (1998) AMS Press. ISBN 0-8218-2130-X
